Letters in Biomathematics Volume I, issue 2 (2014) An International Journal http://www.lettersinbiomath.org

Research Article

Vertical in a Two-Strain Model of Dengue Fever

David Murillo1, Susan A. Holechek1,2,*, Anarina L. Murillo1, Fabio Sanchez1,3, Carlos Castillo-Chavez1

Abstract The role of vertical transmission in vectors has rarely been addressed in the study of dengue dynamics and control, in part because it was not considered a critical population-level factor. In this paper, we apply the pioneering model- ing ideas of Ross and MacDonald, motivated by the context of the 2000–2001 dengue outbreak in Peru, to assess the dynamics of multi-strain competition. An invading strain of dengue (DENV-2) from Asia rapidly circulated into Peru eventually displacing DENV-2 American. A -dengue model that con- siders the competing dynamics of these two DENV-2 genotypes, the resident or the American type and the invasive more virulent Asian strain, is introduced and analyzed. The model incorporates vertical transmission by DENV-2 Asian a potentially advantageous trait. Conditions for competitive exclusion of dengue strains are established. The model is used to show that lower transmission rates of DENV-2 Asian are sufficient for displacing DENV-2 American in the presence of vertical transmission. Keywords: -host model, dengue, , vertical transmission, Peru

 1 Introduction

Sir Ronald Ross (1911) first introduced mathematical models in the study of vector-borne disease dynamics, in the context of Malaria [11, 66]. Over a decade later Kermark and McK- endrick [41] adapted his work in the context of communicable diseases. Today a simplified version of the Kermark McKendrick model is found in most elementary calculus textbooks and is known as the Susceptible-Infected-Recovered or SIR model (Figure 1). The SIR

β IS γ

Figure 1: Simple SIR model. Susceptible, S, infected I, and recovered, R, individuals. Parameter β represents the per capita per infective rate and γ denotes the per capita recovery rate.

1Simon A. Levin Mathematical, Computational, and Modeling Sciences Center, Arizona State University 2Center for Infectious Diseases, Biodesign Institute, Arizona State University 3Escuela de Matematica, Universidad de Costa Rica, San Jose, Costa Rica *Correspondence: [email protected] Vertical Transmission of Dengue Fever Murillo et al. model is given by the following system of nonlinear differential equations:

dS SI dI SI dR dt = −β N , dt = β N − γI, dt = γI. (1)

The basic reproductive number (R0) for System (1) is a dimensionless ratio, R0 = β/γ. The product of the transmission per person rate per unit of time (β) and the average in- fectious period (1/γ), R0 determines whether or not an outbreak will take place depending on whether or not R0 > 1 or R0 < 1. If R0 > 1, then the number of secondary generated by a single infectious individual in a pool of purely susceptible individuals (pro- portion of susceptible in the total population is 1) over his/her average infectious period will be enough for an outbreak. On the other hand, if R0 < 1 then the number of secondary cases generated by a typical infectious individual over his/her mean infectious period will not be enough to sustain growth in the class of infectious individuals and the dis- ease will die out. It is also possible to compute the final epidemic size relationship whenever R0 > 1. For a rescaled system we have that s0 ln = R0(1 − s∞), s∞ where s0 is the proportion of susceptibles at time t = 0 and s∞ is the proportion of suscepti- bles left at the end of the outbreak, that is, at time t = ∞. Since the proportion of infectives (at t = ∞) is equal to zero then by testing for the presence or absence of antibodies to the disease (responsible for the outbreak) prior and after the outbreak, we can estimate R0 from this last relation (for extensions see [35]). The model developed by Ross (1911), now typi- cally referred to as the Ross-MacDonald model, was the first to consider the transmission of a disease that involves a vector and a host. Ross and MacDonald identified via their model the importance of, for example, the host vector ratio, biting rates, and host and vector average infectious periods via the corresponding R0 to the dynamics of malaria. They used their R0 to identify and evaluate control efforts for reducing or eliminating malaria within a host population. This manuscript provides a detailed application of an extension of the Ross-MacDonald model in the context of a disease, dengue, that is creating havoc in Southeast Asia and the Americas. Dengue, a re-emerging vector-borne infectious disease, puts 40% of the global population at risk with 50 to 100 million infections per year, including 500,000 cases of dengue hemorrhagic fever (DHF) and 22,000 deaths, mostly in children [17, 47]. The chal- lenges posed by dengue in the U. S. have attracted increased attention due to global mo- bility [57]. Dengue virus belongs to the genus Flavivirus, family Flaviviridae with four active antigenically distinct serotypes, DENV-1, DENV-2, DENV-3, and DENV-4, [23] and a new DENV-5 serotype, just identified [62]. The pathogenicity of the disease ranges from asymptomatic, mild dengue fever (DF), to dengue hemorrhagic fever (DHF), and dengue shock syndrome (DSS), with children remaining the most affected [32, 60]. DF is charac- terized by a high fever for 3–14 days with possible symptoms including frontal headache, retro-orbital pain, hemorrhagic manifestations, and rash [16]. A more severe and potentially fatal consequence of dengue is DHF, where symptoms may appear after 3–7 days of fever, including severe abdominal pain, persistent vomiting, change in body temperature (fever to hypothermia), hemorrhagic manifestations, or change in mental status [16]. An additional severe outcome of dengue comes from the onset of DSS, marked by early signs of shock, restlessness, cold clammy skin, rapid weak pulse, or narrowing pulse pressure [16]. Section 2 reviews some of what is known about DENV-2 strains, American and Asian; discusses strain- specific and severity and the role of vertical transmission. Section 3 introduces a two-strain model that accounts and explains the role of strain-specific vertical transmission. It also highlights the results of the mathematical analysis and simulations with emphasis on identifying and quantifying necessary, possibly sufficient conditions that guarantee either strain coexistence or competitive exclusion. The role of vertical transmission in providing a competitive advantage of DENV-2 Asian is explored. Section 4, revisits the results of the

– 250 – Letters in Biomathematics manuscript placing them in the context of dengue dynamics in the Americas. Some of the details of the mathematical analysis are collected in appendices.

2 Epidemiology of Dengue in Peru Since 2000

Dengue is a vector borne disease transmitted primarily by the mosquito Ae. aegypti, which has successfully invaded the vast majority of countries in the tropics and sub-tropics [33]. The secondary vector, Ae. albopictus, habituates a considerably larger geographic region with its eggs adapting better to subfreezing temperatures, making it more likely to suc- ceed in areas where it is currently not present like in the U. S. [34, 57]. Vertical trans- mission of dengue fever (transmission via the vectors’ offspring) has been associated with seasonal climate trends and geography (tropics) [2, 7, 40]. Thus, mosquitoes found in high- risk environmental and geographical regions may become more susceptible to and transmission of specific dengue serotypes [7, 44, 74]. Previous studies supported the view that vertical transmission of dengue was insignificant [71, 72] albeit recent findings have shown it not to be negligible in captivity and in the wild for Ae. aegypti and Ae. albopictus [7, 18, 29, 46, 65, 76]. Previous to 2000 only DENV-1 and DENV-2 American genotypes had co-circulated in Peru with neither DHF nor DSS cases reported [45, 56]. The absence of DHF and DSS in Peru prior to 2000, in the presence of co-circulating DENV-1 and DENV-2 American, has been explained, using the data of experiments carried out in laboratories. These studies identified partial cross-immunity conferred by DENV-1 against DENV-2 American but not conferred against the 2000–2001 invading DENV-2 Asian strain [45]. The successful invasion of DENV-2 Asian has been accompanied by subsequent successful invasions of DENV-3 and DENV-4 [56], and so, Peru is now home to at least four dengue serotypes: DENV-1 and DENV-2 American prevalent in its tropical rain forest prior to 2000, DENV-1, DENV-2 (American and Asian), DENV-3, and DENV-4 serotypes co-circulating in its coastal desert areas including Tumbes, Piura, Lambayeque, La Libertad, and Ancash, following the 2000– 2001 outbreak (illustrated in Figure 2). The displacement of DENV-2 American by the DENV-2 Asian genotype has also been associated with the appearance of DHF in Cuba, Jamaica, Venezuela, Colombia, Brazil, and Mexico [27, 50, 54, 63]. Table 1 summarizes factors that play a role in the higher levels of pathogenicity associated with DENV-2 Asian infections while collecting biological evidence supporting the greater virulent strength of DENV-2 Asian. The geographical displacement of DENV-2 American by DENV-2 Asian could be modeled, for example, using complex metapopulation models [11, 12]; and so the question becomes “Can we learn something about dengue displacement dynamics using simpler models?” The rest of the manuscript provides a positive and somewhat satisfactory answer.

2.1 A Brief Overview of Vertical Transmission The standard mode of transmission for dengue is via horizontal transmission (human-to- mosquito). The possibility of vertical transmission (transovarial transmission) has gained attention over the last 30 years due, in part, to advances in molecular biology, that allow us to test for the possibility of transovarial transmission. Vertical transmission provides a possible mechanism supporting virus dengue persistence (in nature) in the absence of a recognized host and/or under unfavorable conditions for mosquito activity [42]. While previous studies have been unsuccessful to demonstrate vertical transmission of dengue [71, 72, 77] other findings have demonstrated that vertical transmission involving Ae. aegypti and Ae. albopictus species is feasible in captivity and in the wild [7, 9, 18, 29, 42, 46, 52, 65, 76]. In field studies, the percentage of natural vertical transmission of DENV from the female to her eggs can be quantified by analyzing the presence of DENV in terms of the minimum infection rates (MIR), which are calculated for vector-bone diseases as the number of (pos- itive pools/total mosquitoes)×1000. The MIR for dengue virus in Ae. aegypti has been

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COLOMBIA ECUADOR Tumbes DENV-1 DENV-2 DENV-3 DENV-4

Piura DENV-1 DENV-2 DENV-3 DENV-4 Lambayeque DENV-1 BRAZIL DENV-2 La Libertad DENV-1 DENV-2

Ancash DENV-3

PACIFIC OCEAN

BOLIVIA

Lima, Capital of Peru CHILE

Figure 2: Distribution of Dengue cases in Peru during 2000–2001 (modified from [56]). Prior to 2000, DENV-1 and DENV-2 American were circulating in the country. The 2000–2001, outbreak introduced DENV-2 Asian, DENV-3 and DENV-4 in Peru increasing the potential risk of DHF and DSS in the country.

Table 1: Selected research evidence that supports DENV-2 Asian virulence. Sev- eral studies demonstrate the pathogenicity and severity of DENV-2 Asian in disease dy- namics. DENV-2 Asian is found to be better equipped for vertical transmission and vec- tor studies have shown that Ae. aegypti are more susceptible to DENV-2 Asian infection. DENV-2 Asian has spread to many countries in South America and the Caribbean Basin. Genetic evidence obtained from mosquito and animal models also highlight the virulence of DENV-2 Asian in these hosts.

Biological Evidence References The presence of intrinsic genetic mutations in DENV-2 Asian [21, 48] The lower rate of successful disseminated infection in Ae. aegypti with DENV-2 [5, 6] American when compared with those associated with DENV-2 Asian Higher virus output in dendritic cells (five-fold difference) within Ae. aegypti [20] mosquitoes for DENV-2 Asian (21%) versus DENV-2 American (3%) Higher vectorial capacity measured by 2 to 65-fold increases in replication of [3] DENV-2 Asian in Ae. aegypti midgut when compared to DENV-2 American DENV-2 Asian is better adapted for vector transmission [5] DENV-1 antibodies can protect against DENV-2 American but not DENV-2 [45] Asian Ae. aegypti is more susceptible to DENV-2 Asian infection [6] American genotype reduces dengue virus output in human monocytes and den- [21] dritic cells DENV-2 Asian takes over Peru, first time with DHF cases [56] DENV-2 Asian takes over the Caribbean Basin [27] Selection of virulent DENV-2 Asian in dendritic cells and mosquitoes [20] Animal model showed higher viremia and rash in DENV-2 Asian versus DENV-2 [58] American DENV-2 Asian has evolved into several genetic lineages co-circulating with other [38] serotypes

– 252 – Letters in Biomathematics calculated in the range of 0.18–5 [2, 6, 48, 55]. Detection of DENV specific serotypes using molecular techniques indicates higher infection rates with DENV-2 in Ae. albopicuts (MIR = 26.6) compared to DENV-2 prevalence in Ae. aegypti (MIR = 0.37 to 1.5) [2, 6, 48] exist, suggesting that Aedes species display different susceptibilities to dengue virus infections. Additional complications are possible, for example, laboratory results from pooled and indi- vidual mosquitoes from regions where dengue is in Peru suggests that Ae. aegypti may be capable of carrying two dengue serotypes simultaneously [14, 49, 51]; a fact that we ignore here, that is, we assume that if possible, its effect is negligible. We assume this because our model focuses on two different strains from the same serotype and replication of DENV-2 Asian in the mosquito vector outcompetes DENV-2 American. Indeed, controlled studies done in a laboratory setting support the hypothesis that higher infection rates exist when Ae. aegypti is exposed to the DENV-2 Asian strain in comparison to DENV-2 Amer- ican. In a study done by Armstrong and Rico-Hesse [6], Ae. aegypti females derived from mosquitoes captured in Texas and Mexico were exposed via oral feeding to DENV-2 Asian viral isolates from different localities including Venezuela (Mara3) and Peru (Iqt2913). The overall rate of disseminated infection in Ae. aegypti was 27–30% for all the DENV-2 Asian isolates versus 9–13% with the American isolates suggesting a higher infection and dissem- ination efficiency of DENV-2 Asian in Ae. aegypti [5] (shown in Figure 3 (left)). Moreover, viral replication in the midgut turned out to be higher in mosquitoes infected with DENV-2 Asian than in those infected with DENV-2 American (shown in Figure 3 (right)) [5]. Despite the results in these studies, the fact remains that there is still no clear way of identifying the biological mechanisms behind the displacement of DENV-2 American by DENV-2 Asian at the population level (the subject of this manuscript). Mathematical and statistical models have indeed added to our overall understanding of dengue dynamics even though most have been inspired or driven by the situation in Asia [61]. Dengue is now endemic in more than 100 countries [79] and yet there is no effective vaccine. Furthermore, the discovery of the new DENV-5 serotype (still undetermined) has naturally raised global concerns [62]. The introduction of DENV-2 Asian in northeastern Peru (2011) was responsible for more than 15,000 suspected dengue cases over three weeks, 93 severe cases, and 14 deaths (mostly children under 15). These data provide some evidence that dengue in the Americas may exhibit significant differences due to host, pathogen, climate or geographic heterogeneity [39]. Clearly, worldwide efforts must also monitor the prevalence of DENV genotypes as well as the dynamics of Ae. aegypti mosquito populations (including males1). Our model considers two different strains from the same serotype and replication of DENV-2 Asian with displacement of DENV-2 American in the population. Here, we are particularly interested in addressing the question of whether or not vertical transmission provides the critical evolutionary advantage needed by DENV-2 Asian to displace DENV-2 American.

3 Two-strain Dengue Model with Vertical Transmission

There have been many attempts to gain insights into the dynamics of vector borne dis- eases (like malaria and dengue). Nishiura (2006) reviews several modeling efforts driven or motivated by dengue. We expand on the simplest framework for dengue presented in [61]. Our model includes progression to DHF but ignores deaths due to DHF (see Figure 4). It also considers two co-circulating genotypes of dengue. Both DHF and DSS, described in the introduction, are severe manifestations of dengue, with associated symptoms being observed following onset of DF (high fever for 3–14 days). DHF and DSS are both thought to be the result of secondary infections with heterologous strains [31, 68]. Our model is based on one specific dengue serotype (DENV-2). While DHF cases are possible in people exposed to one serotype, deaths are low in comparison to DHF cases in a population that

1To estimate the contribution of vertical transmission to the prevalence of dengue in the mosquito pop- ulation.

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Figure 3: The percentages of mosquitoes infected with DENV-2 Asian (Mara3) versus DENV-2 American (Iqt2913) demonstrate that DENV-2 Asian was more prevalent in the infected populations compared to DENV-2 American (left). Differences in the transmis- sion of DENV-2 Asian and DENV-2 American over 14 days in mosquitoes exposed to the respective viral strains is also shown (right). These figures are adapted from the work of Armstrong and Rico-Hesse [5]. has been previously exposed to a heterotypic serotype (in this case DENV-1, DENV-3, and DENV-4). Thus, sequential heterotypic infections involving more than one serotype lead to an increased disease severity by an antibody-dependent enhancement mechanism (ADE) ac- counting for the majority of deaths in patients with DHF/DSS [43, 56, 69]. The population at risk of dengue is roughly 3 billion (40 to 60% of the world population) with 22,000 deaths worldwide [17, 47], which is less than the deaths due to seasonal influenza and pneumonia in the U. S. alone (over 40,000) [37]. Even at this rate, mathematical models have ignored deaths due to influenza and yet still obtain relevant model predictions [36]. Here we adopt a similar approach and ignore deaths due to DHF. We use a SIR model for the host and a SI model for the vector (see [61] for an intro- duction to this framework in the context of Dengue). The host system includes the class of susceptible humans, S, the class of humans infected with American genotype, DAm, the class of humans infected with Asian genotype, DAs, the class of humans that progress to DHF, H, and the class of recovered humans, R, (see Figure 4). Vertical transmission is only possible among vectors infected with DENV-2 Asian (see Figure 5). Vectors act as a reservoir for the transmission of dengue from human to vector and do not recover since they are only carriers of the dengue virus. Thus, given their short lifespan [26], the fact that vectors do not recover supports the concept of vertical transmission, that is, if vectors recovered there would be no significant role for vertical transmission. All of the model parameters are assumed to be (positive) constants. The natural per capita birth/death rate of humans is µ while the rates at which humans become infected with American or Asian genotype from mosquitoes are βAm and βAs, respectively. The per capita progression rates to DHF is α. Further, only individuals infected with DENV-2 Asian can progress to DHF (after presenting DF-like symptoms). The rate at which humans recover from infection from either genotype American, Asian or the extreme DHF stage are δAm, δAs, and δH , respectively. N, the total human population size, is assumed constant since the change in population size is insignificant for a single outbreak, which occurs in less than a year. M, the total vector population size, is assumed constant which is biologically reasonable within a single outbreak since the disease has not shown to reduce the lifespan of the vector in the wild with an average lifespan as an adult of roughly 10 days [26]. Moreover, there are more rigorous mathematical approaches based on quassi-steady approximations that incorporate non-constant population sizes [15] however, this is not within the scope of this manuscript. Individuals diagnosed with DHF are assumed to be hospitalized and so their poten-

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µN µS

β SW β SW M M µD

αD

µD µH δ D δ D

δHH µR

Figure 4: Host model. Susceptible, S, infected with DENV-2 American genotype, DAm, infected with DENV-2 Asian genotype, DAs, progression to DHF, H, and recovered, R, individuals.

µmM - pµmWAs µmV

V

VθAmDAm VθAsDAs N N

WAm WAs pµmWAs

µmWAm µmWAs

Figure 5: Vector model. Susceptible, V, infected with DENV-2 American genotype, WAm, and infected with DENV-2 Asian genotype, WAs. A certain proportion, p, of births confers vertical transmission and enters Was directly.

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Table 2: Default Parameter Values. Biological parameters may vary across geographic and temporal scales, however, most of the values are taken from related literature or esti- mated based on common values.

Parameter Value Definition (Units) References N 1 Total human population size [26] M 1 Total vector population size [26] µ 0.0048 Human natural birth/death rate (per day) Estimated θAs 0.8160 Infectious rate from hosts to vectors, Asian strain (per day) [5] θAm 1.6212 Infectious rate from hosts to vectors, American strain (per day) [5] βAs 0.0483 Infectious rate from vectors to hosts, Asian strain (per day) [21] βAm 0.0182 Infectious rate from vectors to hosts, American strain (per day) [21] α 0.113 Progression rate to DHF from Asian strain (per day) [78] µm 0.0958 Natural mortality rate of vectors (per day) [64] p 0.103 Proportion of vectors infected via vertical transmission [46] δAm 0.2 Recovery rate from, American strain (per day) Estimated δAs 0.2 Recovery rate from, Asian strain (per day) Estimated δH 0.2 Recovery rate from DHF (per day) Estimated tial to spread dengue is considered to be negligible. In our model, the natural per capita birth/death rate of mosquitoes is µm, the rate proportion of mosquitoes that are infected through vertical transmission is p (where p in [0, 1]), the rate at which mosquitoes get in- fected with American genotype is θAm and with Asian genotype is θAs. These parameters and corresponding default values are summarized in Table 2. The system of nonlinear dif- ferential equations used to model the dynamics of two dengue strains in competition is given by β SW β SW S˙ = µN − Am Am − As As − µS, M M β SW D˙ = Am Am − (δ + µ)D , Am M Am Am β SW D˙ = As As − (δ + α + µ)D , As M As As

H˙ = αDAs − (δH + µ)H, (2) R˙ = δAmDAm + δAsDAs + δH H − µR, V θ D V θ D V˙ = µ M − pµ W − Am Am − As As − µ V, m m Am N N m V θ D W˙ = Am Am − µ W , Am N m Am V θ D W˙ = As As + pµ W − µ W . As N m As m As

3.1 Basic Reproductive Number Since M and N are constants, we proceed to rescale the model accordingly, that is, each class is replaced by the proportion of the total respective population. Further, all infected individuals are assumed to recover at the same average per capita rate δ. In the exclusive presence of DENV-2 American, we find that the reproductive number is s Am βAm θAm R0 = · , (δ + µ) µm a dimensionless quantity derived using the next generation operator (see Appendix A). The square root indicates that secondary transmission is a “two-step” process, that is, in order for a human to cause a secondary human infection, a mosquito must be first infected. The average infectious period is 1/(δ + µ), and βAm is the transmission rate (host to vector).

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Am Similarly, θAm is the transmission rate from vectors to hosts. When R0 > 1, we expect an outbreak of dengue in a population facing only DENV-2 American. For the case when DENV-2 Asian is the only active genotype, we find that the repro- ductive number is s 2 As p p βAs θAs R0 = + + · . 2 2 (δ + α + µ) µm

As There are four components to R0 : θAs/µm is the contribution to the reproductive number from infected mosquitoes, the infectious force of mosquitoes times the average time spent in the infectious class; βAs/(δ + α + µ) is likewise the infectious force of humans times the average time spent in the infectious class; (p/2)2 is the indirect contribution from vertical transmission, that is, from infections caused by mosquitoes that were born with dengue; p/2 is the direct contribution from vertical transmission, infectious mosquitoes create more infectious mosquitoes by giving birth to them. Dengue transmission for DENV-2 American is a “two-step” process due to the absence of vertical transmission. The basic reproductive number of a model, that considers simultaneously two competing strains is, by definition, the maximum of the two reproductive numbers

Am As R0 = max R0 , R0 .

In order to assess the role of vertical transmission, we must consider the reproductive number for Asian to be a function of p, that is, s 2 As p p βAs θAs R0 (p) = + + · . (3) 2 2 (δ + α + µ) µm

From equation (3), we can compute the critical vertical transmission value pcrit needed for As Am R0 (p) > R0 given that s As βAs θAs Am R0 (0) = · < R0 . (δ + α + µ) µm

The critical vertical transmission threshold is therefore defined as

Am2 As 2 R0 − R0 (0) pcrit = Am . (4) R0

As Am Therefore R0 (p) > R0 whenever p > pcrit. Hence, from this definition, we see that in the As absence of vertical transmission (p = 0) DENV-2 American would dominate only if R0 (0) < Am R0 . DENV-2 Asian would only dominate (in the absence of vertical transmission) if As As Am R0 (0) > 1 and R0 (0) > R0 . In general, for any p greater than pcrit, if pcrit is in [0, 1] As Am then we have that R0 (p) > R0 , with DENV-2 Asian dominating DENV-2 American As Am As Am thanks to vertical transmission. In general, when R0 (pcrit) = R0 , if R0 (0) ≤ R0 and Am R0 > 1, then for p > pcrit we have that

As As Am R0 (p) > R0 (0) > R0 is the vertical transmission condition needed in order for DENV-2 Asian to overtake DENV-2 American. We describe this in detail in Figure 6 for three cases: (1) pcrit = 0, (2) pcrit = 0.5, As Am and (3) pcrit = 1. In (1), since R0 (0) = R0 (the solid line), then DENV-2 Asian As Am will dominate under any level of vertical transmission since R0 (p) > R0 for all p > 0. As As Am Moreover, if R0 (0) > 1 and R0 (0) > R0 , then DENV-2 Asian simply outcompetes DENV-2 American regardless of whether or not vertical transmission is present. Next, As Am As Am in (2), when R0 (0.5) = R0 (the dotted line), then it is a tie when R0 (0.5) = R0 , but

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As Am any region above the line that is, R0 (p) > R0 , allows DENV-2 Asian to dominate. Lastly, As Am As Am in (3), when R0 (1) = R0 (the dashed line), R0 (0) possibly much less than R0 , when As As Am R0 (p) values above the line are enough for DENV-2 Asian to dominate (R0 (p) > R0 ). In summary, an increase in p leads to a decrease in the threshold required for DENV-2 Asian to outcompete DENV-2 American.

Impact of p on R crit 0 3 p =0 crit p =0.5 2.5 crit p =1 crit 2

(0) 1.5 As R

1

0.5

0 0 0.5 1 1.5 2 2.5 3 R Am

Figure 6: The region where DENV-2 Asian genotype outcompetes DENV-2 American geno- As type is represented by the area above the line for a fixed pcrit. For a fixed R0 (0) and Am R0 , if p is greater than pcrit, graphed, then DENV-2 Asian dominates DENV-2 Ameri- As Am As Am can. Conversely, if p = 0 (the solid line), then R0 (p) > R0 if R0 (0) > R0 . As we As Am increase pcrit to 0.5 (the dotted line), we see that R0 (0) can be less than R0 and still As Am R0 (p) > R0 . If we further increase pcrit to 1 (the dashed line), then DENV-2 American must be much stronger than DENV-2 Asian in order for DENV-2 American to outcompete DENV-2 Asian.

3.2 Equilibria

There are in fact only three equilibria in our system with competitive exclusion preclud- ing strain coexistence. We have a disease free equilibrium (DFE), a DENV-2 American equilibrium (EAm), and a DENV-2 Asian equilibrium (EAs). The DFE is

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ (S = 1,DAm = 0,DAs = 0,H = 0,R = 0,V = 1,WAm = 0,WAs = 0). (5)

The DFE always exists and is locally asymptotically stable whenever R0 < 1, see Ap- pendix A. Further, we make use of a Lyapunov function to show that the DFE is globally asymptotically stable if R0 < 1, see Appendix C. The equations defining EAm are

Am2 µµm R0 − 1 ∗ µβAm + µm(δ + µ) ∗ S = ,DAm = , θAm(βAm + µ) θAm(βAm + µ) ∗ ∗ DAs = 0, H = 0, ∗ ∗ ∗ ∗ ∗ (6) R = 1 − S − DAm, V = 1 − WAm, Am2 µ R0 − 1 ∗ ∗ WAm = ,WAs = 0. Am2 θAmµ R0 + βAm

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Am The EAm only exists if R0 > 1. The equations defining EAs are

∗ θAsµ + µm(δ + α + µ)(1 − p) ∗ S = ,DAm = 0, θAs(βAs + µ) As 2 As 2 ∗ µµmR0 (0) + p − 1 ∗ µµmαR0 (0) + p − 1 DAs = , H = , θAs(βAs + µ) θAs(βAs + µ)(δ + µ) (7) ∗ ∗ ∗ ∗ ∗ ∗ R = 1 − S − DAs − H , V = 1 − WAs, µµ RAs(0)2 + p − 1 W ∗ = 0, W ∗ = m 0 . Am As µRAs(0)2(1 + µm ) + 1 − p 0 βAs

As2 The EAs only exists if R0 (0) > 1.

3.3 Invasion Reproductive Numbers The invasion reproductive number (derived in Appendix B) for DENV-2 Asian invading DENV-2 American is s 2 As 2 As p p R0 (0) Rinv = + + Am . (8) 2 2 R0 Note, if p = 0, then the invasion reproductive number is simply the ratio of the DENV-2 Asian in the absence of vertical transmission to DENV-2 American reproductive numbers, as it would be expected from a simple competition model. In order for EAm to be locally Am asymptotically stable, we must have existence, that is R0 > 1, and strength against inva- As sion, Rinv < 1. The invasion reproductive number associated with the ability of DENV-2 American to invade DENV-2 Asian is

Am Am R0 p Rinv = As (1 − p). (9) R0 (0) We see that if p = 1 then DENV-2 American could never invade DENV-2 Asian. This is because we would always have a reservoir of DENV-2 Asian, a highly unlikely scenario. We As Am have that EAs is locally asymptotically stable whenever R0 > 1 and Rinv < 1. Figure 7a As Am As As shows how Rinv < 1 varies as a function of R0 and R0 (0) for p = 0. Note that Rinv = 1 Am As when R0 = R0 . We observe in Figure 7b (p = 0.5) how the curve has shifted upwards, As making it easier for Rinv > 1. We observe in Figure 7c, (p = 1) how the lowest point of the graph is at 1. To further illustrate the impact of the invasion reproductive number, we will play out a scenario where DENV-2 American is endemic in a population and DENV-2 Asian is the invasive strain. Suppose DENV-2 American is introduced into a dengue-free region and that Am R0 > 1, then we would expect an outbreak. In the long-term, DENV-2 American would become and remain endemic in the population, the situation in Peru prior to 2000. The introduction of DENV-2 Asian into the same (now endemic) region may or may not alter As the role of DENV-2 American. We know that it is not enough to have R0 > 1 to have an outbreak of DENV-2 Asian because not all of the population is in a susceptible state (some are infected with DENV-2 American). Further, there is possibly a segment of the population that has been exposed to dengue and thus it is protected from DENV-2 Asian. As An outbreak of DENV-2 Asian is only possible if we also have that Rinv > 1, the situation in Peru after 2000. This scenario is shown in Figures 8 and 9. We introduce a few infected people with DENV-2 American in a population of mostly susceptible individuals. The DFE Am is unstable because R0 > 1 and the outbreak is dominated by DENV-2 American since Am As R0 > R0 (Figure 8b). DENV-2 American is endemic and DENV-2 Asian dies out in As this scenario (Figure 8a). Then we increase βAs so that Rinv > 1. Another outbreak is triggered, but this time it is dominated by DENV-2 Asian (Figure 9b). The population

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p = 0

2.5

2

1.5 As inv R 1 p = 0.5

0.5

0 3 3

2.5 3 2.5 2 2.5 2 1.5 1.5 1 1 2 0.5

RAm 0.5 0 As inv

As R 0 R 0 1.5

As Am 1 (a) p=0, DENV-2 Asian invades if R0 (0) > R0 .

0.5 3

2.5 3 2 2.5 2 1.5 1.5 1 1 0.5 Am 0.5 R 0 As 0 R 0

(b) Increasing p to 0.5, facilitates DENV-2 Asian to invade as noted by the upward shift of the graph.

p = 1

3

2.8

2.6

2.4

2.2

As inv 2 R 1.8

1.6 RAs versus p inv 1.4

1.2

1 3 3 2.5 3 2 2.5 2 2.5 1.5 1.5 1 1 0.5 2 Am 0.5 R 0 As 0 R 0

As inv 1.5 R (c) As p is further increased to 1, it becomes much 1 easier for DENV-2 Asian to invade. 0.5

0 3

2.5 3 2 2.5 2 1.5 1.5 1 1 0.5 Am 0.5 R 0 As 0 R 0

As (d) Increasing p shifts upward the manifold for Rinv As Am as a function of R0 (0) and R0 . Figure 7: Invasion reproductive number for DENV-2 Asian into DENV-2 American. As p is increased, it becomes easier for DENV-2 Asian to invade DENV-2 American, represented by the upward shift of the graph.

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started near an endemic level of DENV-2 American (EAm), but now the DENV-2 Asian tends to an endemic level (EAs) while DENV-2 American tends to zero (Figure 9a).

RAs=0.98005 RAm=1.2276 inv 0 0.06 0.15

0.1 0.05 Am D 0.05 0.04 0 0 100 200 300 400 500 600 700 800 Time As 0.03 D RAs=1.1908 0

0.15 0.02

0.1 As

D 0.01 0.05

0 0 100 200 300 400 500 600 700 800 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Time D Am (a) First outbreak is dominated by DENV-2 (b) After the initial outbreak, there is an endemic Am As American since R0 > R0 . level of DENV-2 American.

As Am Figure 8: If R0 < 1, R0 > 1, then EAm is stable as DENV-2 American tends to some endemic level and DENV-2 Asian tends to zero (a). The phase portrait shows the relative prevalence of each DENV-2 strain (b). The outbreak starts near the origin and eventually tends towards EAm (square).

RAs=1.5039 RAm=1.2276 inv 0 0.04

0.04 0.035 Am

D 0.02 0.03

0 0.025 0 100 200 300 400 500 Time As 0.02 D RAs=1.8341 0 0.015 0.04 0.01 As

D 0.02 0.005

0 0 100 200 300 400 500 0 0 1 2 3 4 5 6 7 8 Time D −3 Am x 10 (a) Second outbreak is dominated by DENV-2 (b) After the initial outbreak, there is an endemic As Am Asian since R0 > R0 . level of DENV-2 Asian.

As Figure 9: If the outbreaks starts near EAm and Rinv > 1, then DENV-2 American tends to zero and DENV-2 Asian tends to an endemic level (a). The phase portrait shows the relative prevalence of each DENV-2 strain (b). The outbreak starts near EAm (square) and eventually tends towards EAs (circle).

Am Am As Similarly, we can illustrate how Rinv varies as a function of R0 and R0 (0) for fixed Am Am p, see Figure 10. The top most manifold corresponds to p = 0. As R0 increases, Rinv increases. When we increase p, the manifold lowers and flattens out until the point when it is flat at p = 0. At this point, DENV-2 American is no longer able to invade DENV-2 Am Am Asian as no value for R0 will make Rinv > 1

4 Discussion

While Bernoulli was perhaps the first to mathematically model a disease in 1760 [11, 59], it took more than a century before Ross introduced the first models to study the mosquitoes spreading Malaria in the 1900s [11, 28, 70]. Over a century after the groundbreaking work of Ross, models are still being used to study Malaria [53], Dengue [67, 19, 28, 61], and

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RAm versus p inv

4 p=0

p=.33 3

p=.66

Am inv 2 R

1

p=1 0 3 3 2 2.5 2 1 1.5 1 0.5 0 Am 0 R RAs(0) 0 0

Figure 10: As p increases, it becomes more difficult for DENV-2 American to invade DENV-2 Asian, as indicated by the manifold decreasing and flattening out. Ultimately, when p = 1, DENV-2 American is no longer able to invade as represented by the bottom most, flat graph.

Regions of Stability 3

p=0 p=.25 p=.5 2.5 p=.75 II p=1

2 (0) As 0 R

1.5

1

III 0.5 I

0 0 0.5 1 1.5 2 2.5 3 RAm 0

As Figure 11: There are three regions of stability. The DFE is stable when R0 < 1 and Am As2 R0 < 1, Region I. EAs is stable in Region II (when p = 0) if R0 (0) > 1 and As Am R0 (0) > R0 , in this case EAm will be unstable. In Region III, where p>0, EAm is Am As Am stable only if R0 > 1 and R0 (p) < R0 , in this case EAs becomes unstable. However, As2 As Am EAs is stable if R0 (p) > 1 and R0 (p) > R0 , then EAm becomes unstable. You can see that Region II can expand thanks to vertical transmission.

– 262 – Letters in Biomathematics other vector-borne disease such as West Nile Virus [10, 22]. However, the role of vertical transmission in vector-borne models has rarely been studied, but see the contribution by Busenberg and Cooke [13], whereas HIV has become the canonical example of vertical transmission for a sexually-transmitted disease [4, 8, 73]. Similar mathematical models to the one presented here have been developed, [1, 25, 61], suggesting the importance of vertical transmission to the persistence of dengue where there is a widely fluctuating environment and diapause. Our research shows that the landscape is changing by the introduction of a competing strain through vertical transmission into an endemic area, which in turn, makes dengue more likely to both spread in an area without prior infection or invade an area dominated by a “native ” strain. Our model was inspired by the 2000–2001 dengue epidemic in the northwest region of Peru, the first time that DHF cases were observed in this region following the introduction of DENV-2 Asian [56]. Despite the endemic presence of both DENV-1 and DENV-2 American in Peru, no DHF cases were previously detected, possibly explained by the cross-immunity “conferred” by DENV-1 against DENV-2 American strain and not the Asian [45]. Moreover, data from the 2000–2001 outbreak in Peru showed that DENV-2 Asian displaced DENV-2 American as observed by Restriction Fragment Length Polymorphism (RFLP) analysis and corroborated by partial sequence analysis [56]. Displacement of DENV-2 American by the DENV-2 Asian has previously been documented in the Americas [30, 63]. While co-infection with two serotypes is feasible in both the mosquito and human population [14, 49], the fact is that the proportion of individuals reported carrying two or more serotypes has always been insignificant [51]. Moreover, when DENV-2 Asian and DENV-2 American co-circulate DENV-2 Asian would outcompete DENV-2 American [5, 6]. Co-infection with a different serotype should not be ignored as it is biologically possible and it will be explored in a future mathematical model. Invasion reproductive numbers generally arise in the context of two species competing for the same niche leading to competitive exclusion [80]. In an area where neither DENV-2 Asian nor American is present, the basic reproductive number is simply the maximum of the two, strain specific, reproductive numbers

Am As R0 = max R0 , R0 , s Am βAm θAm R0 = · , (δ + µ) µm s 2 As p p βAs θAs R0 (p) = + + · , 2 2 (δ + α + µ) µm

As where we note R0 (p) is a strictly increasing function of the vertical transmission parame- ter, p. In a completely susceptible population, vertical transmission can improve the chances As As of a DENV-2 Asian outbreak (R0 (0) < 1 and R0 (pcrit) > 1 with pcrit in [0, 1]). Further- more, in an area already endemic with DENV-2 American, vertical transmission facilitates the invasion, and replacement by DENV-2 Asian provided the invasion reproductive number is greater than one s 2 As 2 As p p R0 (0) Rinv = + + Am , 2 2 R0 again a situation when vertical transmission can help. Similarly, if DENV-2 Asian is endemic in a region then vertical transmission can make it more difficult for DENV-2 American to invade since Am Am R0 p Rinv = As (1 − p). R0 (0) Our results show that even a low probability of vertical transmission can have a major impact on the long term dynamics of dengue fever. For two competing strains, vertical

– 263 – Vertical Transmission of Dengue Fever Murillo et al. transmission can make a difference between failed outbreaks or invasions, as well as the ability to become endemic in a population. Our model results highlight the importance of vertical transmission in a dengue outbreak as well as the importance of epidemiological surveillance that accounts for molecular genotyping. Routine detection of the virus in both mosquitoes and hosts in endemic areas with dengue will be valuable in order to prevent major outbreaks and gauge the severity of the response that is required to combat any potential outbreak.

Acknowledgements

The authors are solely responsible for the views and opinions expressed in this research; it does not necessarily reflect the ideas and/or opinions of the funding agencies or Arizona State University. This publication was made possible by grant number 1R01GM100471-01 from the National Institute of General Medical Sciences (NIGMS) at the National Institutes of Health. Its contents do not necessarily represent the official views of NIGMS. The authors would also like to acknowledge Kamal Barley for assistance with graphical enhancements.

A R0 via Next Generation Operator

We use the next generation operator to calculate the basic reproductive number. This method has several advantages over other methods especially in the context of vector-borne diseases, see [75] or [24] for a more complete discussion and proof of the method. In this section we outline how this method is applied. First we must identify the infected classes: DAm, DAs, WAm, WAs, and H. Then we must identify the “new” infections. Infections coming into DAm, DAs, WAm, and WAs are new while infections coming into H are “old” since individuals must first be infected via DAs. Then we form two vectors, F which consist of only the new infection terms, and V that is the negation of the remaining terms in our infected classes, that is 2 3 DAm˙ 6 7 6 ˙ 7 6 DAs 7 6 7 6 ˙ 7 F − V = 6WAm7 . 6 7 6 ˙ 7 6 WAs 7 4 5 H˙

Then 2 3 2 3 βAmSWAm M (δ + µ)DAm 6 7 6 7 6 βAsSWAs 7 6 7 6 M 7 6 (δ + α + µ)DAs 7 6 7 6 7 F = 6 θAmVDAm 7 and V = 6 7 . 6 N 7 6 µmWAm 7 6 7 6 7 6 θAsVDAs 7 6 7 6 N + pµmWAs7 6 µmWAs 7 4 5 4 5 0 (δ + µ)H − αDAs

Next we calculate the Jacobian Matrices

2 βAmS 3 2 3 0 0 M 0 0 δ + µ 0 0 0 0 βAsS 6 0 0 0 M 07 6 0 δ + α + µ 0 0 0 7 6 θ V 7 6 7 F = 6 Am 0 0 0 07 , V = 6 0 0 µm 0 0 7 . 6 N 7 6 7 θAmV 0 0 0 µ 0 4 0 N 0 pµm 05 4 m 5 0 0 0 0 0 0 −α 0 0 δ + µ

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Next we must evaluate the Jacobian matrices at the disease free equilibrium (Equation 5), keeping in mind that we have normalized both the human and mosquito populations. Then it only remains to find the eigenvalues of FV−1 since the basic reproductive number is the spectral radius, or largest eigenvalue of the next generation matrix −1 R0 = ρ(FV ). Carrying out the calculation yields two candidates (the other eigenvalues are zero or strictly smaller than these candidates) s s 2 Am βAm θAm As p p βAs θAs R0 = · and R0 = + + · (δ + µ) µm 2 2 (δ + α + µ) µm where Am As R0 = max R0 , R0 as required.

B Invasion Reproductive Numbers

To find the invasion reproductive number, we follow the same methodology as in finding the basic reproductive number. However, instead of assuming the entire population is susceptible to infection, we assume that one strain is already established, endemic, in the population. Then “new” infections are only those infections associated with the invading strain. Then for Asian invading American, only classes DAs, WAs, and H are of interest with new infections in the DAs and WAs classes:

2 βAsSWAs 3 2 3 M (δ + α + µ)DAs F = θAsVDAs and V = µ W . 4 N + pµmWAs5 4 m As 5 0 (δ + µ)H − αDAs Next we calculate the Jacobian Matrices

2 βAsS 3 2 3 0 M 0 δ + α + µ 0 0 F = θAsV and V = 0 µ 0 . 4 N pµm 05 4 m 5 0 0 0 −α 0 δ + µ Next we evaluate these Jacobian matrices at the American equilibrium (Equation 6), since we assumed strain 1 was endemic prior to the introduction of Asian. Then it only remains to find the eigenvalues of FV−1 yields only one candidate v 2 2 u 2 As ! As p up R0 (0) Rinv = + t + . 2 2 Am2 R0

Similarly for American invading Asian, DAm and WAm are the classes of interest. 2 3 2 3 βAmSWAm M (δ + µ)DAm F = 4 5 and V = 4 5 . θAmVDAm N µmWAm Next we calculate the Jacobian Matrices 0 βAmS δ + µ 0 F = M and V = , θAmV 0 µ N 0 m and we evaluate these Jacobian matrices at the Asian equilibrium (Equation 7), since we assumed strain 2 was endemic. Then the dominant eigenvalue of FV−1 is

v 2 u Am2 ! Am u R0 (1 − p) Rinv = t . As 2 R0 (0)

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C Global stability of R0 via the Lyapunov-LaSalle Theorem

Using the rescaled system of equations, let

βAm βAs L = DAm + DAs + WAm + WAs. µm µm Clearly L is positive definite and radially unbounded. The orbital derivative of L is

L˙ = βAmWAmS − (δAm + µ)DAm + βAsWAsS − (δAm + µ)DAm

+ θAmDAmV − µmWAm + θAsDAsV − µmWAs + pµmWAs Am = −βAmWAm(1 − S) − (δAm + µ)DAm(1 − R0 V ) As − βAsWAs(1 − S − pWAs) − (δAs + µ)DAs(1 − R0 (0)V ).

If R0 < 1, then clearly L˙ ≤ 0. Next we must show that the DFE is the maximal invariant subspace of L˙ = 0. From above we have four conditions that must be simultaneously satisfied for L˙ = 0:

WAm(1 − S) = 0, (10)

WAs(1 − S − pWAs) = 0, (11) Am DAm(1 − R0 V ) = 0, (12) As DAs(1 − R0 (0)V ) = 0. (13)

Equation 10 implies that either WAm = 0 or S = 1. Suppose WAm 6= 0, then S = 1 and S˙ = 0 but S˙ = −βAmWAm − βAsWAs < 0, Am which is a contradiction, thus WAm = 0. From Equation 12 either DAm = 0 or V = 1/R0 . Am Suppose DAm 6= 0, since R0 < 1, V > 1 which is impossible since V ∈ [0, 1]. Therefore As DAm = 0. Similarly, Equation 13 implies that either DAs = 0 or V = 1/R0 (0), but if As As R0 < 1, then R0 (0) < 1 and we conclude DAs = 0. Finally, Equation 11 implies that either WAs = 0 or S = 1 − pWAs > 0 if p < 1. Suppose WAs 6= 0. Since DAs = 0, D˙ As = 0, and D˙ As = βAsSWAs > 0, thus we conclude WAs = 0. Also note that since D˙ As = 0, H˙ = −(δH + µ)H which implies H → 0, thus the maximal invariant set is the DFE. Hence, by the Lyapunov-LaSalle Theorem, the DFE is locally asymptotically stable.

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